Local moves on spatial graphs and finite type invariants

2001/06/20 by Kouki Taniyama, Taniyama, Kouki, Akira Yasuhara +1
Mathematics · #57M25 (Primary) 57M15 (Secondary) #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GT #msc:57M15 #msc:57M25

paper · pdf · doi:10.48550/arxiv.math/0106173

LaTeX, 18 pages with figures, to appear in Pacific Journal of Mathematics

arxiv created 2001/06/20 · openalex publication_date 2001/06/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define A_k-moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let A_k-equivalence denote an equivalence relation generated by A_k-moves and ambient isotopy. A_k-equivalence implies A_k-1-equivalence. Let \cal F be an A_k-1-equivalence class of the embeddings of a finite graph into the 3-sphere. Let \cal G be the quotient set of \cal F under A_k-equivalence. We show that the set \cal G forms an abelian group under a certain geometric operation. We define finite type invariants on \cal F of order (n;k). And we show that if any finite type invariant of order (1;k) takes the same value on two elements of \cal F, then they are A_k-equivalent. A_k-move is a generalization of C_k-move defined by K. Habiro. Habiro showed that two oriented knots are the same up to C_k-move and ambient isotopy if and only if any Vassiliev invariant of order ≤ k-1 takes the same value on them. The ` if' part does not hold for two-component links. Our result gives a sufficient condition for spatial graphs to be C_k-equivalent.

Related

• C_k-moves on spatial theta-curves and Vassiliev invariants
• Automorphisms of the 3-sphere that preserve spatial graphs and handlebody-knots
• Topological symmetry groups and mapping class groups for spatial graphs
• Brunnian local moves of knots and Vassiliev invariants
• Finite type invariants for knotoids